Patrick Desrosiers

A note on biorthogonal ensembles

We study multiple orthogonal polynomials in the context of biorthogonal ensembles of random matrices. In these ensembles, the eigenvalue probability density function factorizes into a product of two determinants while the eigenvalue correlation functions can be written as a determinant of a kernel function. We show that the kernel is itself an average of a single ratio of characteristic polynomials. In the same vein, we prove that the type I multiple polynomials can be expressed as an average of the inverse of a characteristic polynomial. We finally introduce a new biorthogonal matrix ensemble, namely the chiral unitary perturbed by a source term, whose multiple polynomials are related to the modified Bessel function of the first kind.

Duality in random matrix ensembles for all β

Abstract Gaussian and Chiral β -Ensembles, which generalise well-known orthogonal ( β = 1 ), unitary ( β = 2 ), and symplectic ( β = 4 ) ensembles of random Hermitian matrices, are considered. Averages are shown to satisfy duality relations like { β , N , n } ⇔ { 4 / β , n , N } for all β > 0 , where N and n respectively denote the number of eigenvalues and products of characteristic polynomials. At the edge of the spectrum, matrix integrals of the Airy (Kontsevich) type are obtained. Consequences on the integral representation of the multiple orthogonal polynomials and the partition function of the formal one-matrix model are also discussed. Proofs rely on the theory of multivariate symmetric polynomials, especially Jack polynomials.

Supersymmetric Calogero–Moser–Sutherland models and Jack superpolynomials

Abstract A new generalization of the Jack polynomials that incorporates fermionic variables is presented. These Jack superpolynomials are constructed as those eigenfunctions of the supersymmetric extension of the trigonometric Calogero–Moser–Sutherland (CMS) model that decomposes triangularly in terms of the symmetric monomial superfunctions. Many explicit examples are displayed. Furthermore, various new results have been obtained for the supersymmetric version of the CMS models: the Lax formulation, the construction of the Dunkl operators and the explicit expressions for the conserved charges. The reformulation of the models in terms of the exchange-operator formalism is a crucial aspect of our analysis.

Hermite and Laguerre β-ensembles: Asymptotic corrections to the eigenvalue density

Abstract We consider Hermite and Laguerre β -ensembles of large N × N random matrices. For all β even, corrections to the limiting global density are obtained, and the limiting density at the soft edge is evaluated. We use the saddle point method on multidimensional integral representations of the density which are based on special realizations of the generalized (multivariate) classical orthogonal polynomials. The corrections to the bulk density are oscillatory terms that depends on β . At the edges, the density can be expressed as a multiple integral of the Konstevich type which constitutes a β -deformation of the Airy function. This allows us to obtain the main contribution to the soft edge density when the spectral parameter tends to ±∞.

Jack Polynomials in Superspace

AbstractThis work initiates the study of orthogonal symmetric polynomials in superspace. Here we present two approaches leading to a family of orthogonal polynomials in superspace that generalize the Jack polynomials. The first approach relies on previous work by the authors in which eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland Hamiltonian were constructed. Orthogonal eigenfunctions are now obtained by diagonalizing the first nontrivial element of a bosonic tower of commuting conserved charges not containing this Hamiltonian. Quite remarkably, the expansion coefficients of these orthogonal eigenfunctions in the supermonomial basis are stable with respect to the number of variables. The second and more direct approach amounts to symmetrize products of non-symmetric Jack polynomials with monomials in the fermionic variables. This time, the orthogonality is inherited from the orthogonality of the non-symmetric Jack polynomials, and the value of the norm is given explicitly.

Asymptotic correlations for Gaussian and Wishart matrices with external source

We consider ensembles of Gaussian (Hermite) and Wishart (Laguerre)

Jack superpolynomials, superpartition ordering and determinantal formulas

N\times N

Classical symmetric functions in superspace

hermitian matrices. We study the effect of finite rank perturbations of these ensembles by a source term. The rank

Orthogonality of Jack polynomials in superspace

r

Generalized Hermite polynomials in superspace as eigenfunctions of the supersymmetric rational CMS model

of the perturbation corresponds to the number of non-null eigenvalues of the source matrix. In the perturbed ensembles, the correlation functions can be written in terms of kernels. We show that for all